When you learn that William Hogarth was a governor of the Foundling Hospital, the strident morality ever-present in his artwork makes a lot of sense. In 18th century London, the theme of progress must have felt pertinent. The hospital, on whose original site the Foundling Museum now stands, started taking in impoverished children in the 1740s. It felt emblematic of everything that was getting bigger or better at the time: philanthropy, industry, a new coffee-addled middle class, and so on.

But then Hogarth made his Rake’s Progress paintings, highlighting the less progressive elements of society by depicting the rise and fall of an unscrupulous wag who gains and squanders an inheritance. It’s an enduring story dense with satire, and modern responses are pluriform — from Igor Stravinsky’s opera with a libretto by WH Auden (yep, that collaboration genuinely happened), to like-for-like artworks such as those displayed in this exhibition.

To mark the 250th anniversary of Hogarth’s death, there are four contemporary reworkings of his masterpiece dotted through the Foundling, in addition to the original Georgian etchings. David Hockney’s Rake’s Progress prints are officially semi-autobiographical, but they also have a very sideways, detached gaze. Created in the early 1960s, they narrate a young man coming into money, then coming to terms with his sexuality in New York. They are personal but also full of big-city bewilderment, and as in Hogarth’s work, the final scene is of destitute madness.

The postcolonial artist Yinka Shonibare explores a story which is contained in the detail of Hogarth’s work, which is the marginalisation of black people. Shonibare’s 1998 work Diary of a Victorian Dandy, which once adorned the London Underground, is a photographic series documenting a day in the life of a debauched dandy. Crucially, the powerful man at the centre is black, not white. The context is later than Hogarth — the imperial era, when class identity became complexified by racial identity.

A new work has been commissioned of the artist Jessie Brennan, who imagines the demolition of a high-rise estate in Tower Hamlets by exhibiting four identical pencil sketches, each more scrunched-up than the last. Portraying ‘what remains of an apparently failed utopian ideal of community housing’, her personal critique of progress is both technically straightforward and ultra-relevant.

Occupying much of the museum basement is Grayson Perry’s set of tapestries created in 2012, called The Vanity of Small Differences. These are, perhaps, the clearest homage to Hogarth, in that acute class-consciousness is their very fibre. Perry’s rake climbs through the strata of modern Britain, his ‘progress’ revealing the shallow obsessions of each layer, most of which ultimately centre on material goods. One figure, the epitome of Middle England, proudly wields a French coffee press, and again we’re staring in the face of Hogarth’s well-to-do, hypocritical middle class. Some things never change — not least the resonance of a strong moral tale.

Progress runs until 7 September at the Foundling Museum, 40 Brunswick Square, London, WC1N 1AZ. Entrance fee included in general admission: £7.50 adults, £5 concessions, free for under-16s, Foundling Friends and Art Fund members, and half-price for National Trust members.